Integrand size = 28, antiderivative size = 165 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x) \log (a+b x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 (a+b x) \log (d+e x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 46} \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^2 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {e^2 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac {e}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {1}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rule 46
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 (d+e x)} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{b^2 (b d-a e) (a+b x)^3}-\frac {e}{b^2 (b d-a e)^2 (a+b x)^2}+\frac {e^2}{b^2 (b d-a e)^3 (a+b x)}-\frac {e^3}{b^3 (b d-a e)^3 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {e}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x) \log (a+b x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 (a+b x) \log (d+e x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.56 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-((b d-a e) (-3 a e+b (d-2 e x)))+2 e^2 (a+b x)^2 \log (a+b x)-2 e^2 (a+b x)^2 \log (d+e x)}{2 (b d-a e)^3 (a+b x) \sqrt {(a+b x)^2}} \]
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Time = 2.86 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {\left (2 \ln \left (b x +a \right ) b^{2} e^{2} x^{2}-2 \ln \left (e x +d \right ) b^{2} e^{2} x^{2}+4 \ln \left (b x +a \right ) x a b \,e^{2}-4 \ln \left (e x +d \right ) x a b \,e^{2}+2 \ln \left (b x +a \right ) a^{2} e^{2}-2 \ln \left (e x +d \right ) a^{2} e^{2}-2 x a b \,e^{2}+2 b^{2} d e x -3 a^{2} e^{2}+4 a b d e -b^{2} d^{2}\right ) \left (b x +a \right )}{2 \left (a e -b d \right )^{3} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(156\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {b e x}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}+\frac {3 a e -b d}{2 a^{2} e^{2}-4 a b d e +2 b^{2} d^{2}}\right )}{\left (b x +a \right )^{3}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{2} \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{2} \ln \left (b x +a \right )}{\left (b x +a \right ) \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\) | \(213\) |
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Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (119) = 238\).
Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {b^{2} d^{2} - 4 \, a b d e + 3 \, a^{2} e^{2} - 2 \, {\left (b^{2} d e - a b e^{2}\right )} x - 2 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{3} d^{3} - 3 \, a^{3} b^{2} d^{2} e + 3 \, a^{4} b d e^{2} - a^{5} e^{3} + {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} x^{2} + 2 \, {\left (a b^{4} d^{3} - 3 \, a^{2} b^{3} d^{2} e + 3 \, a^{3} b^{2} d e^{2} - a^{4} b e^{3}\right )} x\right )}} \]
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\[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {b e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )} - \frac {e^{3} \log \left ({\left | e x + d \right |}\right )}{b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{2} d^{2} - 4 \, a b d e + 3 \, a^{2} e^{2} - 2 \, {\left (b^{2} d e - a b e^{2}\right )} x}{2 \, {\left (b d - a e\right )}^{3} {\left (b x + a\right )}^{2} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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